## Introduction

The S2LET code provides high performance routines for fast wavelet analysis of signals on the sphere. It uses the SSHT code
built on the MW
sampling theorem to
perform exact spherical harmonic transforms on the sphere. The SO3 code is also needed for the exact Wigner transform on the rotation group. The
resulting wavelet transform is theoretically exact, i.e. a band-limited signal can be recovered from its wavelet coefficients exactly and the wavelet coefficients capture all the information. **S2LET** also supports the HEALPix sampling scheme, in which case the transforms are not theoretically exact but achieve good numerical accuracy.

**S2LET** supports scale-discretised axisymmetric wavelets (Wiaux et al 2008, Leisted et al 2012), directional wavelets (McEwen et al 2015), ridgelets (McEwen 2015), and curvelets (Chan et al 2017), as well as needlets (Marinucci et al 2008, Baldi et al 2006) and B-spline wavelets (Starck et al 2006). It exploits a fast Wigner transform on the rotation group (implemented in SO3) to perform the scale-discretised wavelet transform, which anyalyses of scalar and spin signals are both supported.

This page outlines the main features of the code, installation details as well as the core functionalties and interfaces. References, version, and license information then follows.
**S2LET** requires the SSHT, SO3, and FFTW libraries. The IO FITS features require CFITSIO. To support HEALPix, a valid installation of its Fortran implementation must be provided. More details about an installation from scratch can be found on the Dependencies page.

## Scale discretised wavelets on the sphere

In **S2LET**, the scale-discretised wavelets are constructed through an exact tiling of harmonic space, following the scale-discretised approach described in Wiaux et al (2008). The harmonic line is tiled into wavelet kernels which are localised (i.e. have compact support) in both real and frequency spaces, as shown on the image below for a particular set of wavelet parameters (B=2, J_min=2, harmonic space on the left, corresponding kernels on the right).

With an exact spherical harmonic transform as the one provided in SSHT, the wavelet transform implemented in **S2LET** is theoretically exact in both pixel and harmonic spaces. In other words one can decompose a band-limited signal in a set of wavelet maps that exactly capture all the information, and reconstruct the initial signal at floating-point precision. Due to the nature of the tiling in harmonic space, the individual wavelet maps have different band-limits and hence can be reconstructed at different resolutions, i.e. with the minimal number of samples. The left and right panels below are the wavelet decomposition of Earth tomography data at full (left) and multi-resolution (right). Due to the exactness of all transforms, these two approaches are equivalent and capture all the information contained in the initial band-limited map.

When using HEALPix instead of the MW sampling on the sphere, the harmonic transform is not exact and hence the corresponding wavelet transform performed by **S2LET** is only exact in harmonic space. In other words, a set of spherical harmonic coefficients may be projected onto wavelets exactly in harmonic spaces but the HEALPix maps don't capture all the information contained in these coefficients. However good numerical accuracy is achieved in a wide range of situations.

## Needlets and B-spline wavelets

In addition to scale-discretised wavelets (Wiaux et al 2008), **S2LET** supports Needlets (Marinucci et al 2008, Baldi et al 2006) and B-spline wavelets (Starck et al 2006). Please refer to these papers for the **S2LET** paper for the details of the construction of these wavelets. Importantly, in **S2LET** they are also constructed through a tiling or a windowing of the harmonic line. The figure on the left illustrates the differences between the three constructions and the resulting support of the axisymmetric wavelets. Note that the three types of wavelets work with the MW sampling theorem and Healpix.

In terms of usage, **S2LET** has a unique interface for the wavelet transform to avoid duplication of high-level routines. The default setting uses scale-discretised wavelets, and the wavelet type can be changed with dedicated functions. Thus one can write programs and pipelines with the high-level routines and decide on the type of wavelets later on.

## Spin, directional wavelets

The framework of scale-discretised wavelets was extended in McEwen et al (2015) to the directional setting (following Wiaux et al 2008) and spin signals. The figure below shows the wavelet decomposition of the Earth topography using 3 directions.

## Ridgelets

**S2LET** supports the spherical ridgelet transform developed in McEwen (2015). The ridgelet transform is defined natively on the sphere, probes signal content globally along great circles, does not exhibit any blocking artefactes, does not rely on any ad hoc parameters, and permits the exact inversion for antipodal signals.
The figure below shows spherical ridgelets, with axis aligned with the North pole, for wavelet scales j=3 and 4, plotted on the sphere and parmetrically.

## Curvelets

**S2LET** also supports scale-discretised curvelets
developed in Chan (2017)), which are efficient to represent local linear and
curvilinear strcuture and admit exact inversion for both scalar and spin signals.
The figure below shows spherical curvelets,
with axis aligned with the North pole, for various wavelet scales j, plotted on the sphere.

## Installation and documentation

The core functionalities of **S2LET** are written in C and are self-documenting. The core C library only requires SSHT and FFTW. The extra IO and HEALPix features require the CFITSIO and HEALPix libraries. Interfaces are provided for all high-level routines in Matlab, IDL and Java. In Matlab and IDL, these interfaces allow one to read/write FITS maps (for both the MW and HEALPix formats), compute the spherical harmonic and the wavelet transforms and plot the resulting signals on the sphere. To support HEALPix, **S2LET** uses a hybrid C/Fortran interface to the Fortran HEALPix library. Several examples, tests and demos are provided fo the C library and all interfaces in Matlab, IDL and Java.

#### Compiling

If you need to install the dependencies required by **S2LET** please visit the Dependencies page.

The instuctions and options to build the main C library are detailed on this page.

Note that we provide a makefile as well as a Cmake configuration file to facilitate cross-platform compilation.

Instuctions to build and use the Matlab interfaces are detailed here.

Instuctions to build and use the IDL interfaces are detailed here.

#### Source code documentation

**S2LET** ships with source and HTML documentation.

- The C documentation is generated by doxygen and available here.

- The Matlab routines that interface with the C implementation are self
documenting, and documentation can be accessed through the help command in
Matlab). HTML documentation is available here and is built with m2html.

- The documentation for IDL interfaces is available here and can be rebuilt in IDL with the function s2let_make_doc. The DOC_LIBRARY routine can also be used to display the documentation while running IDL.

## Download

We make the source code of the **S2LET** package
available under the license described below.

**S2LET** can be downloaded from the following site:

## Referencing

If you use S2LET for work that results in publication, please reference this site (http://www.s2let.org/) and the related academic papers:

J. D. McEwen, B. Leistedt, M. Büttner, H. V. Peiris, Y. Wiaux,Directional spin wavelets on the sphere, IEEE Trans. Signal Proc., submitted, 2015 (ArXiv).

J. D. McEwen, M. Büttner, B. Leistedt, H. V. Peiris, Y. Wiaux,A novel sampling theorem on the rotation group, IEEE Sig. Proc. Let., 22(12):2425-2429, 2015 (ArXiv | DOI).

B. Leistedt, J. D. McEwen, P. Vandergheynst and Y. Wiaux,S2LET: A code to perform fast wavelet analysis on the sphere, Astronomy & Astrophysics, 558(A128):1-9, 2013 (ArXiv | DOI).

J. D. McEwen,Ridgelet transform on the sphere, IEEE Trans. Signal Proc., submitted, 2015 (ArXiv).

J. Y. H. Chan, B. Leistedt, T. D. Kitching, J. D. McEwen,Second-generation curvelets on the sphere, IEEE Trans. Signal Proc., 65(1):5-14, 2017 (ArXiv | DOI).

If the MW sampling is used please also cite the following paper:

J. D. McEwen and Y. Wiaux, A novel sampling theorem on the sphere, IEEE Trans. Signal Proc., 59, 5876-5887, 2011 (ArXiv | DOI).

Note that the theoretical framework for scale-discretised wavelets is detailed in this work:

Y. Wiaux, J. D. McEwen, P. Vandergheynst, O. Blanc, Exact reconstruction with directional wavelets on the sphere, Mon. Not. Roy. Astron. Soc., 388(2):770-788, 2008. (ArXiv | DOI).

Needlets are detailed in these papers:

Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D. 2009, Annals of Statistics, 37/3, 1150.

Marinucci, D., Pietrobon, D., Balbi, A., et al. 2008, Mon. Not. Roy. Astron. Soc., 383, 539.

B-spline wavelets were first introduced in this paper:

Starck, J.-L., Moudden, Y., Abrial, P., & Nguyen, M. 2006a, Astron. & Astrophys., 446, 1191.

The Healpix scheme is detailed in

Gorski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759.

## License

S2LET package to perform fast wavelet transform on the sphere

Copyright (C) 2012 Boris Leistedt & Jason McEwenThis program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details (LICENSE.txt).

You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

## Authors and contributors

S2LET was developed by Boris Leistedt, Martin Büttner, Jennifer Chan, and Jason McEwen at University College London (UCL).